Symmetry:
Without knowledge of symmetry we would need incredible many lines connecting the atoms of a
structure to describe a crystal.
Symmetry is the property of an object to be indistinguishable from its initial appearance after some
real or imaginary movements.
Rotation by 180°
Symmetry operations:
There are operations (movements) that lead to superimposition of an object on itself. That means
the object looks the same afterwards. For this purpose we can define symmetry elements, imaginary
geometric entities like lines, points or planes about which the operations take place.
Among these movements we can find some that leave at least one point of the object in a fixed
position. We call them point symmetry operations. They need symmetry elements that intersect in
one point within the object (gravity center).
Point symmetry operations:

m – reflection through a symmetry plane:
 A left handed object is converted into a right handed one.
 Mirror plane = xz-plane:
Every point (x, y, z) is reflected to a position (x, -y, z)
Examples:
Fly (bilaterale Symmetrie), cup, hands, ammonium (3 mirror planes)

i – inversion through an inversion center:
1. Move the point along a straight line to the inversion center.
2. Elongate the line to double length and move the point to its end.
 Inversion center = the origin of the coordinate system:
Every point (x, y, z) is translated to a position (-x, -y, -z)
!
This is an improper rotation about an 2-fold axis.
Examples:
benzol, ethane

Cn – rotation about an n-fold rotation axis (by 360°/n)
 The structure will be unchanged afterwards.
 2-fold rotation axis along z-axis:
Every point (x, y, z) is rotated to a position (-x, -y, z)
Order (n)
2
Minimum rotation in degrees
180
Symbol
21
3
4
6
43 means 3*360°/4 = 270°
120
90
60
31, 32
41, 42, 43
61, 62, 63, 64, 65
Examples:
Water molecule (n = 2), Ethane (n = 2 or 3)
3. Sn – improper rotation about an improper n-fold axis:
1. Rotation by 360°/n
2. Reflection through a plane perpendicular to rotation axis
Examples:
Ethane
Point symmetry operations  point groups:
Point symmetry operations can be combined in 32 ways:
- Crystallography: 32 crystal classes
- Spectroscopy: 32 point groups
 They can be applied to finite objects.
Infinite objects: two or three dimensional lattices
!
Some operations are allowed no more.
!
We need to introduce a new operation.
!
We have to define a crystal lattice.
Crystal lattice:
The fundamental characteristic of a crystal is its high degree of internal order: In it an object
(asymmetric unit) is arranged in a regular way and this arrangement (unit cell) is repeated in all
directions. Therefore you can regard it as a infinite repeating array. The pattern of repetition we call
lattice type:
Square lattice: Sitting in the unit cell you see the same stuff after every rotation about 90°.
Rectangular lattice: Sitting in the unit cell you see the same stuff after every rotation about 180°.
Hexagonal lattice: Sitting in the unit cell you see the same stuff after every rotation about 60°.
Unit cells have to fill the surface in 2D or the space in 3D. You can imagine them stacked like boxes.
Not allowed point symmetry operations:
Only 2-, 3-, 4-, 6-fold rotation axis are allowed in 2D/3D lattices. For this reason you cannot fill the
space with pentagons, heptagons, octagons etc.
Do not be irritated by virus envelops because they lack a lattice:
A new symmetry operation:
For repetition you need to define translation:
*
*
*
*
*
*
*
*
*
*
*
*
You can regard a lattice as a rule of translation.
-
 We now can introduce new symmetry elements:
Screw axis:
1. Rotation around an n-fold axis
2. Translation by 1/n of the unit cell dimension (a, b or c) parallel to the axis
Exampel:
31 screw axis parallel to the c-axis of unit cell = rotation by 120° followed by a displacement
of c/3
-
Glide mirror plane:
1. Reflection through a plane
2. Translation by ½ of the unit cell dimension (a, b, c or n = diagonal) parallel to the glide
plane
Crystal systems and Bravais lattices:
A parallelelepiped needs three vectors (a, b, c) forming its edges defined by three angles (, β, ).
There are 6 crystal systems:
 Cubic
 Tetragonal
 Orthorhombic
 Monoclinic
 Triclinic
 Hexagonal
Each of the 32 crystal classes belongs to one crystal system. I.e. all crystal classes of the cubic system
posses four 3-fold axis proceeding from corner to corner through the center of the unit cell. Crystal
classes of the hexagonal system possess a single 6-fold axis.
There are three lattice types:




Primitive (P):
one point at each corner
-> one lattice point per unit cell
Body centered (I):
one point at each corner + one point at the center
-> 2 lattice points per unit cell
Face centered (F):
one point at each corner + points at the centers of either just one pair of faces or all three
pairs of faces
-> 4 lattice points per unit cell
Base centered (C):
one point at each corner + points at the centers of just one pair of faces
The combination of the crystal classes with the lattice types creates 14 lattice types established by
Auguste Bravais (1811-1863):
Triclinic/anorthic
No restriction
P
1
Monoclinic
 =  = 90°
P, C
2
Orthorhombic
 = β =  = 90°
P, C, I, F
4
Tetragonal
 = β =  = 90°, a = b
P, I
2
Trigonal/hexagonal
 = β = 90°,  = 120°, a = b
P
1
Rhombohedral
 = β = , a = b = c
P
1
Cubic
 = β =  = 90°, a = b = c
P, I, F
3
14
Space groups:
Combine the available symmetry operations (which are the point groups (crystal classes) together
with glide mirror plane and screw axis) with the 14 Bravais lattices yielding 230 combinations.
This 230 space groups are described in international tables of crystallography.
Example: P212121 is a primitive lattice with three times a 2-fold rotation axis.
!
Remember that proteins, RNA and DNA are chiral (C and ribosal C). Centric operations
would change their chirality:
o No reflection
o No inversion (remember inversion = reflection + rotation by 180° about an
orthogonal axis)
 This reduces the possible groups to 65 chiral/enantiomeric space groups.
Most common among proteins are P212121 and P21.
Point symmetry operations:
- reflection
- rotation
- inversion
- improper rotation
32 crystal classes (= point groups)
Member of
Lattice types:
- Primitive
- Base centered
- Face centered
- Body centered
Introducing translation:
- screw axis
- glide mirror plane
Crystal systems:
- cubic
- tetragonal
- orthorhombic
- monoclinic
- triclinic
- hexagonal
14 Bravais lattices
230 Space groups
Unit cell and asymmetric unit:
The unit cell is the smallest unit generating the entire crystal by means of translation:
-
It has the maximum symmetry.
Often you can recognize more than one repeating units in a crystal.
It is not necessary that the unit cell contains a complete object.
The unit cell can but does not have to show intrinsic symmetry.
The asymmetric unit (the motif) is the minimum unit generating the unit cell and thereby the
crystal by means of symmetry operations:
-
It lacks symmetry.
Space group P1: The whole unit cell is the asymmetric unit.
Other space groups than P1: The asymmetric unit is 1/Z of the unit cell. Z is the number of
equivalent positions.
Often there are 4 molecules per unit cell and one per asymmetric unit.
Example – twofold axis and mirror plane => 4 equivalent positions:
P212121 => Z = 4
-
The volume of the asymmetric unit is V(unit cell)/Z.
The solvent content of the asymmetric unit:
P(sol) = 100%*(V(asy)-nV(mol))/V(asy)
n is the number of molecules in the asymmetric unit.
V(mol) = [(M(mol)/(1.35g/ml))/NA]*1024Å3 with the molecular weight of the molecule
M(mol), the density of the protein 1.35g/ml and the Avogadro number
NA=6.022*10-23particle/mol
Rule of thumb: The weight of one amino acid is about 110Da. So M(mol)  110Da * x with x is the
number of residues.
Crystal packing:
 salt bridges and H-bonds weaker than hydrophobic interaction holding proteins
Therefore protein crystals are fragile and proteins within them are probably not significantly
distorted from their native conformation.
Matthews volume:
VM = V(unit cell)/(M(mol)*Z*n)
n is the number of molecules in the asymmetric unit (often 2).
Z is the number of asymmetric units.
 VM = 1.7-4.0Å3/Da (most observed values: 2.15-2.4Å3/Da
A higher Matthew volume means that there is more than one molecule in the
asymmetric unit.